Chapter 4: Q1E (page 157)

Question: Find a general solution to the given differential equation.
$2 y'' + 7 y' - 4 y = 0$

Short Answer

Expert verified

Answer

The general solution of the given equation is ${y = c_{1} e^{\frac{1}{2} t} + c_{2} e^{- 4 t} .}_{}$

Step by step solution

Write the auxiliary equation of the given differential equation.

The given differential equation is $2 y'' + 7 y' - 4 y = 0 .$ $2 y'' + 7 y' - 4 y = 0 .$

The auxiliary equation for the above equation is $2 m^{2} + 7 m - 4 = 0 .$

Find the roots of the auxiliary equation.

Solve the auxiliary equation,

$2 m^{2} + 7 m - 4 = 0 2 m^{2} + 8 m - m - 4 = 0 2 m (m + 4) - 1 (m + 4) = 0 (m + 4) (2 m - 1) = 0$

The roots of the auxiliary equation are $m_{1} = \frac{1}{2}, & m_{2} = - 4 .$

When an auxiliary equation has distinct real roots as&, then the general solution is given as;

$y = c_{1} e^{m_{1} t} + c_{2} e^{m_{2} t}$

the general solution of the given equation is $y = c_{1} e^{\frac{1}{2} t} + c_{2} e^{- 4 t} .$

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Question: Find a general solution to the given differential equation.
$2 y'' + 7 y' - 4 y = 0$

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Find the roots of the auxiliary equation.

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Question: Find a general solution to the given differential equation.2y''+7y'-4y=0

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Find the roots of the auxiliary equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Question: Find a general solution to the given differential equation.
$2 y'' + 7 y' - 4 y = 0$